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We study smooth isotopy classes of complex curves in complex surfaces from the perspective of the theory of bridge trisections, with a special focus on curves in [Formula: see text] and [Formula: see text]. We are especially interested in bridge trisections and trisections that are as simple as possible, which we call efficient. We show that any curve in [Formula: see text] or [Formula: see text] admits an efficient bridge trisection. Because bridge trisections and trisections are nicely related via branched covering operations, we are able to give many examples of complex surfaces that admit efficient trisections. Among these are hypersurfaces in [Formula: see text], the elliptic surfaces [Formula: see text], the Horikawa surfaces [Formula: see text], and complete intersections of hypersurfaces in [Formula: see text]. As a corollary, we observe that, in many cases, manifolds that are homeomorphic but not diffeomorphic have the same trisection genus, which is consistent with the conjecture that trisection genus is additive under connected sum. We give many trisection diagrams to illustrate our examples.

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We give a new, elementary proof that Khovanov homology with Z/2Z-coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that δ-graded knot Floer homology is mutation-invariant. Using the Clifford module structure on HFK induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let L ′ be a link obtained from L by mutating the tangle T . Suppose some rational closure of T corresponding to the mutation is the unlink on any number of components. Then L and L ′ have isomorphic δ-graded HFK groups over Z/2Z as well as isomorphic Khovanov homology over Q. We apply these results to establish mutation-invariance for the infinite families of Kinoshita-Terasaka and Conway knots. Finally, we give sufficient conditions for a general Khovanov-Floer theory to be mutation-invariant.2010 Mathematics Subject Classification. 57M27; 57R58.The complex (CKh(D), d Kh ) possess two gradings, the quantum and homological grading, and the differential preserves the quantum grading and increases the homological grading by one. Thus the homology Kh(D) splits into the direct sum of bigraded modules Kh i,j (D) where i denotes the homological grading and j the quantum grading.Let p be a fixed basepoint in the plane contained in the diagram D. The basepoint p determines a chain map X p : CKh(D) → CKh(D) that squares to 0. The kernel of X p is a subcomplex and reduced Khovanov homology is its homology, with a shift in the quantum grading:It is an invariant of L up to isotopies supported away from p. While over Z the reduced homology depends on the component containing the basepoint, this is not true over Z/2Z.for every link L and any basepoint p. In particular, Kh(L) is well-defined independent of p.Consequently, when discussing reduced Khovanov homology over Z/2Z we will suppress any mention of the basepoint p.Khovanov homology satisfies Kunneth-type formulas for disjoint unions and connected sums. The following properties are well-known and the proofs are easy deductions from the definition of Khovanov homology and Proposition 2.1.Lemma 2.2. Let L 1 , L 2 be arbitrary links. Over Z/2Z there are isomorphismsfor any choice of connected sum.Khovanov homology satisfies an unoriented skein exact triangle. Fix a crossing of D and let D 0 , D 1 denote the 0-and 1-resolutions of D at this crossing. Since Khovanov homology is computed from a cube of resolutions, the complex CKh(D) is, up to a grading shift, the mapping cone of a chain map

This paper introduces two constructions of Legendrian submanifolds of P × R, called Legendrian products and spinning, and computes their classical invariants, the Thurston-Bennequin invariant and the Maslov class. These constructions take two Legendrians K, L and returns a product K × L, they generalize other previous constructions in contact topology, such as frontspinning and hypercube tori, and are equivalent in R 2n+1 . Interestingly, this construction relies upon the explicit embeddings of K, L and not their Legendrian isotopy class.